Convergence in distribution of order statistic random variables with uniform distribution.

Question

I am trying to find the asymptotic distribution of an order statistic $X_{(1)}$ for iid RVs $X_1, ..., X_n \sim \mathrm{Unif}(0,1)$,

The distribution for $X_{(1)}$

$F_{X_{(1)}}(x) = 1 - \left(1 - {x}\right)^{n}, \quad y>0$.

why $X_{(1)}$ convergence in distribution to zero?

I think $F_{X_{(1)}}(x)$ go to $1$ as $n$ go to infinity.

Answer 1

$F_{X_1} (x)=0$ if $x \leq 0$. $F_{X_1} (x) \to 1$ for $ x> 0$. So the limiting CDF is exactly the CDF of the zero random variable.